Gaussian hypergeometric functions and cyclotomic matrices

Let $q=p^n$ be an odd prime power and let ${\mathbb{F}}_q$ be the finite field with $q$ elements. Let $\widehat{{\mathbb{F}}_q^{\times }}$ be the group of all multiplicative characters of ${\mathbb{F}}_q$ and let $\chi$ be a generator of $\widehat{{\mathbb{F}}_q^{\times }}$. In this paper, we investigate arithmetic properties of certain cyclotomic matrices involving nonzero squares over ${\mathbb{F}}_q$. For example, let $s_1,s_2,\cdots ,s_{(q-1)\mathrm{/}2}$ be all nonzero squares over ${\mathbb{F}}_q$. For any integer $1\le r\le q-2$, define the matrix $$B_{q,2}({\chi }^r):={[{\chi }^r(s_i+s_j)+{\chi }^r(s_i-s_j)]}_{1\le i,j\le (q-1)\mathrm{/}2}\mathrm{.}$$ We prove that if $q\equiv 3(\mathrm{m}\mathrm{o}\mathrm{d}4)$, then $$\det (B_{q,2}({\chi }^r))=\prod _{0\le k\le (q-3)\mathrm{/}2}{J}_q({\chi }^r,{\chi }^{2k})=\{\begin{array}{ll}(-1{)}^{\frac{q-3}{4}}{\mathbf{i}}^n{G}_q({\chi }^r{)}^{\frac{q-1}{2}}\mathrm{/}\sqrt{q}& ifr\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}2),\\ G_q({\chi }^r{)}^{\frac{q-1}{2}}\mathrm{/}q& ifr\equiv 0(\mathrm{m}\mathrm{o}\mathrm{d}2),\end{array}$$ where $J_q({\chi }^r,{\chi }^{2k})$ and $G_q({\chi }^r)$ are the Jacobi sum and the Gauss sum over ${\mathbb{F}}_q$ respectively.